Section 31: Problem 3 Solution
Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
Show that every order topology is regular.
Let
where
or
can be infinite. Let
be a point in
if there is such a point, otherwise
(note that in this case
, therefore, no points below
are limit points of
). Similarly choose
in
or
if the interval is empty. Then
and
.